Rathematical Sciences: Nonlinear Problems of Solid Mechanics

该项目将涉及一些非线性的研究 固体力学中的问题。 这些问题包括：（1） 粘弹性、耗散和冲击的动力学问题 （包括非线性粘弹性的一维问题） 体和自持振荡，霍普夫分叉，和 流体-固体相互作用）;（2）光柔性的渐近性 （3）非线性弹塑性分叉问题; (4)接触问题;（5）各向异性中的共存相 介质;（6）其他问题，如逆分叉 问题，非线性磁弹性和控制问题。 每个问题都将给出一个精确的公式（没有特别的 几何简化），每个问题都采用一般 满足标准不变性的本构方程 条件和轻微的身体限制。 使用这种 本构方程阐明的数学结构， 控制方程，并允许检测新的影响， 与物质反应有关。 研究的目标是 发现新的现象，特别是那些与 不稳定性和阈值的发生，以处理混凝土 阐明一般理论的问题，并发展 处理一般和具体问题的分析工具 问题 各种动态和稳态非线性问题， 将研究杆、壳和三维固体。 这些物体是由非线性弹性的，粘弹性的， 塑料或磁致弹性材料。 在每种情况下， 不变的，几何精确的理论，包括一般 将使用非线性本构方程。 的目标 这些研究旨在发现新的非线性效应， 本构方程阈值定性分离 不同的反应，检查重要类型的不稳定性， 有助于冲击和耗散机制的理论， 固体，并开发新的非线性分析方法， 固体力学问题。

This project will involve research on a number of nonlinear problems in solid mechanics. These problems include: (1) dynamical problems of viscoelasticity and dissipation and shocks (including one-dimensional problems for nonlinearly viscoelastic bodies and self-sustained oscillations, Hopf bifurcations, and fluid-solid interactions); (2) asymptotics of light flexible bodies; (3) bifurcation problems of nonlinear elastoplasticity; (4) contact problems; (5) coexistence of phases in anisotropic media; and (6) other problems such as inverse bifurcation problems, nonlinear magnetoelasticity, and control problems. Each problem will be given an exact formulation (without ad hoc geometric simplifications) and each problem employs general constitutive equations satisfying the standard invariance conditions and mild physical restrictions. The use of such constitutive equations illuminates the mathematical structure of the governing equations and permits the detection of new effects associated with material response. The goals of the research are to detect new phenomena, especially those associated with instabilities and the occurrence of thresholds, to treat concrete problems that illuminate the general theories, and to develop analytic tools for the treatment of both general and specific problems. A variety of dynamic and steady-state nonlinear problems for rods, shells, and three-dimensional solid bodies will be studied. The bodies are composed of nonlinearly elastic, viscoelastic, plastic, or magnetoelastic materials. In each case, fully invariant, geometrically exact theories encompassing general nonlinear constitutive equations are to be used. The goals of these studies are to discover new nonlinear effects, determine thresholds in constitutive equations separating qualitatively different responses, examine important kinds of instabilities, contribute to the theory of shocks and dissipative mechanisms in solids, and develop new methods of nonlinear analysis for problems of solid mechanics.